Natural Morita equivalence, extended Green theory, and defect subpairs

Abstract

Let G be a finite group and let $({\mathcal {O}}, K, k={\mathcal {O}}/J({\mathcal {O}}))$ be a p-modular system so that ${\mathcal {O}}$ is a complete valuation ring with fraction field K and residue field $k={\mathcal {O}}/J({\mathcal {O}})$ an algebraically closed field of characteristic p as in finite group modular representation theory. Let A be a finitely generated ${\mathcal {O}}$-free ${\mathcal {O}}$-algebra such that A has a simple ${\mathcal {O}}$-subalgebra S with ${\textsf{1}}_{A} \in S$ and (consequently) A is naturally Morita equivalent to $C_{A}(S)$. We prove that an extension of the Green theory is “compatible” with this Morita equivalence. Then we apply these results to the situation in which G stabilizes a defect subpair (i.e., where (Q, f) is a subpair such that Z(Q) is a defect group of $f \in Bl({\mathcal {O}}C_{G}(Q))$. These analyses are applied to “Weights of Blocks” and “Blocks with a Normal Defect Group”.